Inference for Quantitative Data: Means
t-distributions and one-sample inference
When σ is unknown (always in practice), use the t-distribution with df = n − 1. It is symmetric and bell-shaped with heavier tails than normal, becoming normal as n grows. The one-sample t-statistic is t = (x̄ − μ₀)/(s/√n), and the confidence interval is x̄ ± t*·s/√n. Conditions: random sample, independence (10% condition), and normality. The normality condition is assessed differently by sample size - small n demands a roughly symmetric, outlier-free plot; large n (roughly n ≥ 30) tolerates moderate skew because of the Central Limit Theorem.
Two-sample and paired designs
For two independent means, use the difference x̄₁ − x̄₂ with standard error √(s₁²/n₁ + s₂²/n₂) and conservative df = min(n₁ − 1, n₂ − 1) (or technology). For paired data, analyze the differences and run a one-sample t-procedure on them - never treat paired observations as independent groups. A frequent trap: failing to recognize a matched-pairs design in a pre/post study.
Interpreting results
Interpret a confidence interval as an estimate of the true mean difference in context; interpret a test conclusion in terms of evidence about μ. Do not say a test "proves" the alternative, and do not claim a significant result is practically important - significance depends on n and effect size together.
Worked example. A sample of 16 gives x̄ = 52, s = 8. Testing H₀: μ = 50 vs. H₁: μ > 50 yields t = 2/(8/4) = 1.0 with df = 15. The p-value is roughly 0.17, so there is insufficient evidence to reject H₀ at α = 0.05.